Mathematics for the interested outsider

We continue our project to show that the spaces are actually Banach spaces with Minkowski’s inequality. This will allow us to conclude that is a normed vector space. It states that if and are both in , then their sum is in , and we have the inequality

We start by considering Hölder’s inequality in a toy space I’ll whip up right now. Take two isolated points, and let each one have measure ; the whole space of both points has measure . A function is just an assignment of a pair of real values , and integration just means adding them together. Hölder’s inequality for this space tells us that

where and are Hölder-conjugate to each other. We can set , , and and use this inequality to find

Dividing out and raising both sides to the th power, we conclude that . Thus if both and are integrable, then so is . Thus must be in .

Now we calculate

Dividing out by we find that

This lets us conclude that is a vector space. But we can also verify the triangle identity now. Indeed, if , , and are all in , then Minkowski’s inequality shows us that

which is exactly the triangle inequality we want. Thus is a norm, and is a normed vector space.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.